About Standard Deviation Calculator
Calculate standard deviation, variance, mean, and other statistical measures for your data set.
Supports both sample and population calculations.
What is Standard Deviation?
Standard deviation (σ) measures the amount of variation or dispersion in a set of values. A low
standard deviation indicates values are close to the mean, while a high standard deviation indicates
values are spread out over a wider range.
Sample vs Population
| Aspect |
Sample |
Population |
| Definition |
Subset of data |
Complete data set |
| Symbol |
s |
σ |
| Variance Formula |
Σ(x - x̄)² / (n - 1) |
Σ(x - μ)² / n |
| When to Use |
Estimating from sample |
Have all data points |
Formulas
Sample Standard Deviation
s = √[Σ(x - x̄)² / (n - 1)]
Where:
x̄ = sample mean
n = sample size
Σ = sum of
Population Standard Deviation
σ = √[Σ(x - μ)² / n]
Where:
μ = population mean
n = population size
Other Statistics
- Mean: μ = Σx / n
- Variance: σ² = Σ(x - μ)² / n
- Median: Middle value when sorted
- Mode: Most frequently occurring value
- Range: Maximum - Minimum
- IQR: Q3 - Q1
Example Calculation
Data: 2, 4, 4, 4, 5, 5, 7, 9
Step 1: Calculate mean
μ = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
Step 2: Calculate deviations
(2-5)²=9, (4-5)²=1, (4-5)²=1, (4-5)²=1, (5-5)²=0, (5-5)²=0, (7-5)²=4, (9-5)²=16
Step 3: Sum squared deviations
Σ(x-μ)² = 9+1+1+1+0+0+4+16 = 32
Step 4: Calculate variance (sample)
s² = 32 / (8-1) = 32 / 7 = 4.571
Step 5: Calculate standard deviation
s = √4.571 = 2.138
Interpreting Results
- Low SD (< 1): Data points are very close to the mean
- Medium SD (1-3): Moderate spread around the mean
- High SD (> 3): Data points are widely spread
- SD = 0: All values are identical
Common Applications
- Finance: Measure investment risk and volatility
- Quality Control: Monitor process consistency
- Research: Analyze experimental data variability
- Education: Compare test score distributions
- Weather: Predict temperature variations
Tips
- Use sample standard deviation (n-1) when working with a subset of data
- Use population standard deviation (n) when you have all possible data points
- Outliers can significantly affect standard deviation
- Always check your data for entry errors before calculating
- Standard deviation has the same units as your original data
- A coefficient of variation above 100% indicates high variability
Understanding the Results
Standard Deviation (σ or s)
Measures spread of data. Approximately 68% of data falls within 1 SD of the mean, 95% within 2 SD,
and 99.7% within 3 SD (for normal distributions).
Variance (σ² or s²)
Square of standard deviation. Less intuitive but important for statistical calculations.
Coefficient of Variation (CV)
Expresses standard deviation as a percentage of the mean. Useful for comparing variability between
different data sets.
Interquartile Range (IQR)
Difference between 75th and 25th percentiles. More resistant to outliers than standard deviation.
Mean Absolute Deviation (MAD)
Average of absolute deviations from the mean. Alternative measure of spread.
Standard Error (SE)
Estimate of how much sample mean varies from population mean. SE = SD / √n
When to Use Which Statistic
| Scenario |
Use This |
Reason |
| Measuring spread |
Standard Deviation |
Most common measure |
| Comparing different units |
Coefficient of Variation |
Unit-free comparison |
| Data with outliers |
IQR or MAD |
Less sensitive to extremes |
| Center of data |
Mean or Median |
Mean for normal, median for skewed |
| Most common value |
Mode |
Useful for categorical data |
Common Mistakes to Avoid
- Using population formula for sample data (underestimates variability)
- Ignoring outliers that may skew results
- Comparing SD values from different scales
- Assuming low SD always means better (context matters)
- Forgetting to check if data is normally distributed
Real-World Examples
Example 1: Test Scores
Class A: 85, 87, 88, 90, 92 (Mean: 88.4, SD: 2.7)
Class B: 70, 80, 90, 95, 100 (Mean: 87, SD: 12.1)
Both classes have similar means, but Class A is more consistent while Class B shows more variation
in performance.
Example 2: Manufacturing
Target: Bottle should contain 500ml ± 5ml
Sample: 498, 501, 499, 502, 500, 497, 503
Mean: 500ml
SD: 2.16ml
Process is well-controlled since SD is small and all values within tolerance.
Example 3: Investment Returns
Stock A: Returns: 5%, 6%, 7%, 6%, 5% (SD: 0.7%)
Stock B: Returns: -2%, 8%, 12%, 3%, -1% (SD: 5.8%)
Stock A is less risky (lower volatility) while Stock B has higher risk/reward.
Statistical Distributions
Normal Distribution (Bell Curve)
- 68% of data within ±1 SD from mean
- 95% of data within ±2 SD from mean
- 99.7% of data within ±3 SD from mean
Skewed Distributions
- Right-skewed: Mean > Median (long tail to right)
- Left-skewed: Mean < Median (long tail to left)
- SD still valid but median may be better center measure
Advanced Concepts
Pooled Standard Deviation
Combines standard deviations from two or more groups for comparison.
Coefficient of Variation
CV = (SD / Mean) × 100%. Useful for comparing variability of different data sets with different units
or scales.
Z-Score
Z = (x - μ) / σ. Tells you how many standard deviations a value is from the mean.
Frequently Asked Questions
Why use n-1 for sample standard deviation?
Using n-1 (Bessel's correction) provides an unbiased estimate of population variance. Using n would
systematically underestimate the true population variance.
Can standard deviation be negative?
No. Since it's calculated from squared deviations, it's always zero or positive. Zero means all
values are identical.
What's the difference between variance and standard deviation?
Variance is SD squared. Standard deviation is preferred because it's in the same units as the
original data, making it easier to interpret.
Is a higher standard deviation always bad?
Not necessarily. It depends on context. In quality control, lower is better. In investments, it
indicates higher risk but potentially higher returns.
How do outliers affect standard deviation?
Outliers significantly increase standard deviation since deviations are squared. Consider using
median and IQR if outliers are present.
What sample size do I need?
Minimum of 30 is often recommended for reliable estimates. Larger samples provide more accurate
estimates of population parameters.